Implicit sampling for parameter estimation

Abstract:

Implicit sampling is a Monte Carlo (MC) method that has been shown to be efficient for online state estimation and filtering problems. Here we explain how to use implicit sampling for parameter estimation, i.e. to find independent samples of a posterior probability density. The basic idea is to focus the computational effort on the region of high probability defined by a prior and data. This region is located via numerical optimization, and explored by samples that are generated by solving algebraic equations with a stochastic right hand side. We also present an efficient numerical implementation of implicit sampling for parameter estimation. Specifically, we use BFGS optimization coupled to an adjoint code, and explain how to use multiple grids (coarse to fine) to speed up the computations. The random maps we use to solve the random algebraic equations are pre-conditioned with information from a Hessian. We demonstrate the applicability and efficiency of our method in numerical experiments where we estimate a diffusion coefficient in an elliptic equation, as is important in reservoir simulation and pollution modeling. Our numerical tests indicate that our implementation of implicit sampling can be more efficient than standard MC sampling by a large factor, and that it accounts in full for the nonlinearity of the elliptic inverse problem.